Base vectors in covariance and contravariance are defined through the gradient of coordinate functions, where covariant base vectors are normal to these functions and tangent vectors represent the coordinate lines. The confusion arises from understanding how these definitions apply in practice, particularly in the context of the relationships between the base vectors and their magnitudes. The covariant base vector, expressed as e(1) = ∇u, is derived from the normal vector definition, while the tangent vectors are defined by their orientation along the coordinate lines. Clarification is sought on the mathematical relationships, such as the dot products and their implications for orthogonality and magnitude. Understanding these definitions and relationships is crucial for grasping the concepts of covariance and contravariance in vector spaces.
